But what does it mean?

To most of us, it’s “the number in the middle” or a number that is “balanced”. I’m a fan of taking multipleviewpoints, so here’s another interpretation of the average:

The average is the value that can replace every existing item, and have the same result. If I could throw away my data and replace it with one “average” value, what would it be?

One goal of the average is to understand a data set by getting a “representative” sample. But the calculation depends on how the items in the group interact. Let’s take a look.

The Arithmetic Mean

The arithmetic mean is the most common type of average:

Let’s say you weigh 150 lbs, and are in an elevator with a 100lb kid and 350lb walrus. What’s the average weight?

The real question is “If you replaced this merry group with 3 identical people and want the same load in the elevator, what should each clone weigh?”

In this case, we’d swap in three people weighing 200 lbs each [(150 + 100 + 350)/3], and nobody would be the wiser.

Pros:

It works well for lists that are simply combined (added) together.

Easy to calculate: just add and divide.

It’s intuitive — it’s the number “in the middle”, pulled up by large values and brought down by smaller ones.

Cons:

The average can be skewed by outliers — it doesn’t deal well with wildly varying samples. The average of 100, 200 and -300 is 0, which is misleading.

The arithmetic mean works great 80% of the time; many quantities are added together. Unfortunately, there’s always those 20% of situations where the average doesn’t quite fit.

Median

The median is “the item in the middle”. But doesn’t the average (arithmetic mean) imply the same thing? What gives?

Humor me for a second: what’s the “middle” of these numbers?

1, 2, 3, 4, 100

Well, 3 is the middle of the list. And although the average (22) is somewhere in the “middle”, 22 doesn’t really represent the distribution. We’re more likely to get a number closer to 3 than to 22. The average has been pulled up by 100, an outlier.

The median solves this problem by taking the number in the middle of a sorted list. If there’s two middle numbers (even number of items), just take their average. Outliers like 100 only tug the median along one item in the sorted list, instead of making a drastic change: the median of 1 2 3 4 is 2.5.

Pros:

Handles outliers well — often the most accurate representation of a group

Splits data into two groups, each with the same number of items

Cons:

Can be harder to calculate: you need to sort the list first

Not as well-known; when you say “median”, people may think you mean “average”

Some jokes run along the lines of “Half of all drivers are below average. Scary, isn’t it?”. But really, in your head, you know they should be saying “half of all drivers are below median“.

Figures like housing prices and incomes are often given in terms of the median, since we want an idea of the middle of the pack. Bill Gates earning a few billion extra one year might bump up the average income, but it isn’t relevant to how a regular person’s wage changed. We aren’t interested in “adding” incomes or house prices together — we just want to find the middle one.

Again, the type of average to use depends on how the data is used.

Mode

The mode sounds strange, but it just means take a vote. And sometimes a vote, not a calculation, is the best way to get a representative sample of what people want.

Let’s say you’re throwing a party and need to pick a day (1 is Monday and 7 is Sunday). The “best” day would be the option that satisfies the most people: an average may not make sense. (“Bob likes Friday and Alice likes Sunday? Saturday it is!”).

Similarly, colors, movie preferences and much more can be measured with numbers. But again, the ideal choice may be the mode, not the average: the “average” color or “average” movie could be… unsatisfactory (Rambo meets Pride and Prejudice).

Pros:

Works well for exclusive voting situations (this choice or that one; no compromise)

Gives a choice that the most people wanted (whereas the average can give a choice that nobody wanted).

Simple to understand

Cons:

Requires more effort to compute (have to tally up the votes)

“Winner takes all” — there’s no middle path

The term “mode” isn’t that common, but now you know what button to look for when playing around with your favorite statistics program.

Geometric Mean

The “average item” depends on how we use our existing elements. Most of the time, items are added together and the arithmetic mean works fine. But sometimes we need to do more. When dealing with investments, area and volume, we don’t add factors, we multiply them.

Let’s try an example. Which portfolio do you prefer, i.e. which has a better typical year?

Portfolio A: +10%, -10%, +10%, -10%

Portfolio B: +30%, -30%, +30%, -30%

They look pretty similar. Our everyday average (arithmetic mean) tells us they’re both rollercoasters, but should average out to zero profit or loss. And maybe B is better because it seems to gain more in the good years. Right?

Wrongo! Talk like that will get you burned on the stock market: investment returns are multiplied, not added! We can’t be all willy-nilly and use the arithmetic mean — we need to find the actual rate of return:

Portfolio A:

Return: 1.1 * .9 * 1.1 * .9 = .98 (2% loss)

Year-over-year average: (.98)^(1/4) = 0.5% loss per year (this happens to be about 2%/4 because the numbers are small).

Portfolio B:

1.3 * .7 * 1.3 * .7 = .83 (17% loss)

Year-over-year average: (.83)^(1/4) = 4.6% loss per year.

A 2% vs 17% loss? That’s a huge difference! I’d stay away from both portfolios, but would choose A if forced. We can’t just add and divide the returns — that’s not how exponential growth works.

Some more examples:

Inflation rates: You have inflation of 1%, 2%, and 10%. What was the average inflation during that time? (1.01 * 1.02 * 1.10)^(1/3) = 4.3%

Coupons: You have coupons for 50%, 25% and 35% off. Assuming you can use them all, what’s the average discount? (i.e. What coupon could be used 3 times?). (.5 * .75 * .65)^(1/3) = 37.5%. Think of coupons as a “negative” return — for the store, anyway.

Area: You have a plot of land 40 × 60 yards. What’s the “average” side — i.e., how large would the corresponding square be? (40 * 60)^(0.5) = 49 yards.

I’m sure you can find many more examples: the geometric mean finds the “typical element” when items are multiplied together. You take a set of numbers, multiply them, and take the Nth root (where N is the number of items you're considering).

I had wondered for a long time why the geometric mean was useful — now we know.

Harmonic Mean

The harmonic mean is more difficult to visualize, but is still useful. (By the way, “harmonics” refer to numbers like 1/2, 1/3 — 1 over anything, really.) The harmonic mean helps us calculate average rates when several items are working together. Let’s take a look.

If I have a rate of 30 mph, it means I get some result (going 30 miles) for every input (driving 1 hour). When averaging the impact of multiple rates (X & Y), you need to think about outputs and inputs, not the raw numbers.

average rate = total output/total input

If we put both X and Y on a project, each doing the same amount of work, what is the average rate? Suppose X is 30 mph and Y is 60 mph. If we have them do similar tasks (drive a mile), the reasoning is:

X takes 1/X time (1 mile = 1/30 hour)

Y takes 1/Y time (1 mile = 1/60 hour)

Combining inputs and outputs we get:

Total output: 2 miles (X and Y each contribute “1″)

Total input: 1/X + 1/Y (each takes a different amount of time; imagine a relay race)

And the average rate, output/input, is:

If we had 3 items in the mix (X, Y and Z) the average rate would be:

It’s nice to have this shortcut instead of doing the algebra each time — even finding the average of 5 rates isn’t so bad. With our example, we went to work at 30mph and came back at 60mph. To find the average speed, we just use the formula.

But don’t we need to know how far work is? Nope! No matter how long the route is, X and Y have the same output; that is, we go R miles at speed X, and another R miles at speed Y. The average speed is the same as going 1 mile at speed X and 1 mile at speed Y:

It makes sense for the average to be skewed towards the slower speed (closer to 30 than 60). After all, we spend twice as much time going 30mph than 60mph: if work is 60 miles away, it’s 2 hours there and 1 hour back.

Key idea: The harmonic mean is used when two rates contribute to the same workload. Each rate is in a relay race and contributing the same amount to the output. For example, we’re doing a round trip to work and back. Half the result (distance traveled) is from the first rate (30mph), and the other half is from the second rate (60mph).

The gotcha: Remember that the average is a single element that replaces every element. In our example, we drive 40mph on the way there (instead of 30) and drive 40 mph on the way back (instead of 60). It’s important to remember that we need to replace each “stage” with the average rate.

A few examples:

Data transmission: We’re sending data between a client and server. The client sends data at 10 gigabytes/dollar, and the server receives at 20 gigabytes/dollar. What’s the average cost? Well, we average 2 / (1/10 + 1/20) = 13.3 gigabytes/dollar for each part. That is, we could swap the client & server for two machines that cost 13.3 gb/dollar. Because data is both sent and received (each part doing “half the job”), our true rate is 13.3 / 2 = 6.65 gb/dollar.

Machine productivity: We’ve got a machine that needs to prep and finish parts. When prepping, it runs at 25 widgets/hour. When finishing, it runs at 10 widgets/hour. What’s the overall rate? Well, it averages 2 / (1/25 + 1/10) = 14.28 widgets/hour for each stage. That is, the existing times could be replaced with two phases running at 14.28 widgets/hour for the same effect. Since a part goes through both phases, the machine completes 14.28/2 = 7.14 widgets/hour.

Buying stocks. Suppose you buy $1000 worth of stocks each month, no matter the price (dollar cost averaging). You pay $25/share in Jan, $30/share in Feb, and $35/share in March. What was the average price paid? It is 3 / (1/25 + 1/30 + 1/35) = $29.43 (since you bought more at the lower price, and less at the more expensive one). And you have $3000 / 29.43 = 101.94 shares. The “workload” is a bit abstract — it’s turning dollars into shares. Some months use more dollars to buy a share than others, and in this case a high rate is bad.

Again, the harmonic mean helps measure rates working together on the same result.

Yikes, that was tricky

The harmonic mean is tricky: if you have separate machines running at 10 parts/hour and 20 parts/hour, then your average really is 15 parts/hour since each machine is independent and you are adding the capabilities. In that case, the arithmetic mean works just fine.

Sometimes it’s good to double-check to make sure the math works out. In the machine example, we claim to produce 7.14 widgets/hour. Ok, how long would it take to make 7.14 widgets?

Prepping: 7.14 / 25 = .29 hours

Finishing: 7.14 / 10 = .71 hours

And yes, .29 + .71 = 1, so the numbers work out: it does take 1 hour to make 7.14 widgets. When in doubt, try running a few examples to make sure your average rate really is what you calculated.

Conclusion

Even a simple idea like the average has many uses — there are more uses we haven’t covered (center of gravity, weighted averages, expected value). The key point is this:

The “average item” can be seen as the item that could replace all the others

The type of average depends on how existing items are used (Added? Multiplied? Used as rates? Used as exclusive choices?)

It surprised me how useful and varied the different types of averages were for analyzing data. Happy math.

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115 Comments on "How To Analyze Data Using the Average"

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Prateek Sharma

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Awesome post!!

When I was in school I crammed these formulas to pass the exam because no teacher would satisfy my curiosity behind the why, what & how of it. My math teacher would try to explain it to me but his jargon was most of the time out of comprehension.. My dad later helped me grasp the stuff.

But I must say you did a superb job of putting the concept in a super easy language and indeed there is no one else could have better explained!

I wish there were more teachers like you.. God bless you!

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9 years 5 months ago

adamsmith

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“Let’s say you weigh 160 lbs, and are in an elevator with a 100lb kid and 350lb walrus. What’s the average weight?”

“In this case, we’d swap in three people weighing 200 lbs each [(150 + 100 + 350)/3], and nobody would be the wiser.”

In the first part I quoted, you put the wrong number in. Just thought I’d be nitpicky.

That’s an interesting way to think about the average; I guess I always knew about that, but I’d never explicitly thought about the average being “replace everything with identical things”.

I love your articles; always make me think about things in a different way.

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@Prateek: Wow, thanks for the kind words! Glad you are finding the site useful. I’m glad your dad was able to help you out — sometimes you just need to get things from a different viewpoint.

Unfortunately, math is one of those subjects where topics get one (and only one) explanation, and you’re off to the next one.

@Zac: Happy you’re enjoying the site — I fixed up the typo [I had actually put in my own weight instead of the hypothetical numbers which are easier to add up :) ].

Yeah, it’s amazing how many things we’ve “learned” in the past but haven’t seen from all angles (there’s a few other cool interpretations of the average but I didn’t want the post to be too long). Glad you’re enjoying the articles.

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That my friend is one very well put together article! Thank you for the effort taken to show to us ‘simple’ people, how fun math can be, esp; statistics!

Now if you can just get the fudge heads at Oracle and Microsoft to introduce this into their ‘superior’ databases, we will all have much more straight forward lives indeed ;-)

10/10

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9 years 5 months ago

Kat

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This will come in handy for math homework, cause math is my absolute WORST subject. lol…..

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[…] ・How To Analyze Data Using the Average […]

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9 years 5 months ago

Dave

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This is absolutely brilliant! Knowledge is kind of like comedy. You’ve got to have delivery. If the delivery sucks the response will probably not be very good either. This my friend was fabulous! I would love to study under an individual such as yourself.

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Quite delightful and informative. You do the world a service by contributing to a global increase in knowledge of mathematics.

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[…] How To Analyze Data Using the Average | BetterExplained (tags: statistics mathematics math analysis business education maths reference) […]

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9 years 5 months ago

NebulousMaker

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Informative, Concise, what other everyday uses do you have for old school stat info? How about some information about how we can use calculus in everyday life?

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@mrhassell: Thanks, glad you found it useful! The funny thing is I’m one of those simple people too — I want things to be simple and clear instead of rigorous and opaque. Unfortunately, I’m pretty powerless to influence the db designers :).

@kat: Cool, hope it comes in handy.

@Dave: Appreciate the comment, and I totally agree — any subject can be interesting if presented in the right way. I’ll keep cranking out the “aha!” moments as they happen :).

@Chris: Thanks, glad you found it useful. Yeah, one of the great things about blogging is that everyone can add a bit of information into the world.

@NebulousMaker: Thanks, a series on calculus is on the way. It’s a tricky subject to cover with real, everyday applications (i.e., non-physics), but they’re definitely out there.

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[…] How To Analyze Data Using the Average | BetterExplained (tags: education mathematics analysis) […]

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[…] How To Analyze Data Using the Average | BetterExplained – You drove to work at 30 mph, and drove back at 60 mph. What was your average speed? Hint: It’s not 45 mph, and it doesn’t matter how far your commute is. Read on to understand the many uses of this statistical tool. […]

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[…] How To Analyze Data Using the Average | BetterExplained (tags: math statistics average mean median mode) […]

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A couple more things I noticed after a (second? third? fourth? millionth? I lost count) readthrough (sorry in advance if I’m too notpicky):

“The average is the value that can replace every existing item, and have the same result.” That doesn’t quite apply for the mode or median; for the rest it works, but the median and mode are different ideas entirely. They have their uses, yes, but they don’t fit under that definition of the average.

This might be nitpicky, but I don’t really see where this is analysing data using the average. It shows how to find the average and explains why that works, but it doesn’t really say anything about using that to analyze the data. To me, analyzing data has more to do with variance and the standard deviation, but that might just be me. Maybe I just don’t see it.

Also, here’s something else, possibly more useful for the average, though maybe a little too in depth for this article. The median works to eliminate outliers, but it’s more effective (though more time consuming) to find the mean AFTER eliminating the outliers. Gives a better indicator of what the average really is.

Just a few thoughts on the article and possibly something to put in another article (if I’m too nitpicky, just tell me).

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Hi Zac, no worries at all, I like hearing what works and what doesn’t for each article — it’s a good way to improve.

Yeah, on second thought the title may be misleading — it’s more about “understanding the types of averages, with examples” vs “data analysis”, which probably deserves its own follow-up article. The idea of throwing away the outliers is a good one, and helps clean up data that may otherwise be skewed.

I had a similar inkling about the term “average” being applied to the median and mode — hopefully it’s clear that those items aren’t really “averages” so a replacement doesn’t work. But I’ll think about ways to clarify the sentence.

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9 years 4 months ago

MS

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Excellent post. I tried doing a similar explanation a few months ago, so I have a real appreciation for your(better)work.

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Hi MS, thanks for the comment, glad you enjoyed it. It can be a tricky concept to get across, but I’d still encourage you to trackback / release your explanation also — everyone has different insights :).

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[…] How To Analyze Data Using the Average @ BetterExplainedThe average is a simple term with several meanings. The type of average to use depends on whether you’re adding, multiplying, grouping or dividing work among the items in your set. […]

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To Zac’s point, I just realized the median and mode behave “as if all items had the same value”, in a way.

When choosing the mode (most popular), you are acting as if every value was the mode — it’s the only one that matters. The median is similar: you choose a middle value, and the median doesn’t change if you had replaced every value with it.

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9 years 4 months ago

rakesh juyal

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Oh.. I can understand! yawn

whts the answer anyway:

Quick quiz: You drove to work at 30 mph, and drove back at 60 mph. What was your average speed?

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Hi Rakesh, just check out the section on the harmonic mean.

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9 years 4 months ago

Mohammad Ali

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Dear Kalid, great post once again although I what I got confused into was why do we multiply to get average return of a portfolio over four years? Why dont we add? Similarly what is the logic behind multiplying the diferent rates of inflation across the three periods and not adding them? Can you please clarify a bit on the geometric mean?